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What does this mean? $$\Large.\overline9 $$ I've never seen this notation before.

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    $\begingroup$ Perhaps $0.9999...$? $\endgroup$
    – ajotatxe
    Jun 2, 2015 at 23:28
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    $\begingroup$ It also means $1$, by that's a whole 'nother discussion. $\endgroup$ Jun 3, 2015 at 0:32
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    $\begingroup$ @AlbertMasclans Apparently this notation style depends on geography (en.wikipedia.org/wiki/Repeating_decimal): overline=US, overdot=China, parentheses=Europe. $\endgroup$ Jun 3, 2015 at 8:02
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    $\begingroup$ @Nordik Yes, it depends on geography, but parentheses are used only sometimes in Europe. In Germany we typically use the overline, too. Parentheses are used to denote an uncertanty in the last digit(s). $\endgroup$
    – Christoph
    Jun 3, 2015 at 10:58
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    $\begingroup$ By the way, the overline is technically known as a vinculum. $\endgroup$ Jun 3, 2015 at 14:38

5 Answers 5

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It is called a vinculum and it denotes a repeating decimal.

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    $\begingroup$ +1 for a good word I need to remember to use when the time is right. $\endgroup$ Jun 2, 2015 at 23:58
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    $\begingroup$ Sounds like something in a gynecologist's office. $\endgroup$
    – zhw.
    Jun 3, 2015 at 0:36
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    $\begingroup$ @JoeTaxpayer: Do you mean +1 or +0.9...? $\endgroup$
    – mipadi
    Jun 3, 2015 at 22:18
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It means a repeating decimal. One can write $\frac 16=0.1\overline 6$, or $\frac 1{14}=0.0\overline{714852}$ for example. The repeating part is whatever is under the overline.

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As other answers have said, it stands for a repeating decimal, where the digits under the line are repeated. $$0.\overline{9} = 0.9999999\ldots$$

But if you want to be a little pedantic, you might prefer to say that both $0.\overline{9}$ and $0.9999999\ldots$ are two different forms of notation for the same number. That number is the limit of the sequence formed by repeatedly appending copies the digits covered by the line. In other words, given the notation $0.\overline{9}$, you can write out the following sequence: $$\begin{align} a_1 &= 0.9 \\ a_2 &= 0.99 \\ a_3 &= 0.999 \\ a_4 &= 0.9999 \\ &\vdots \end{align}$$ As you tack on more and more repetitions, these numbers get closer and closer to some limiting value, which I'll call $A$. If you know calculus notation, $$\lim_{n\to\infty} a_n = A$$ The number represented by $0.\overline{9}$ is $A$. It happens to work out to be $1$ (or if we're being pedantic, $1$ is yet another notation for the same number).

As another example of this way of interpreting repeating decimals, consider $0.257\overline{143}$. You can write the sequence $$\begin{align} b_1 &= 0.257143 \\ b_2 &= 0.257143143 \\ b_3 &= 0.257143143143 \\ b_4 &= 0.257143143143143 \\ &\vdots \end{align}$$ and similarly, the number represented by $0.257\overline{143}$ is the value that this sequence gets closer and closer to as you add more repetitions; or $$\lim_{n\to\infty} b_n$$ This one works out to $\frac{128443}{499500}$.

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  • $\begingroup$ Did you just randomly pick that last fraction or did you already have it in mind? $\endgroup$ Jun 3, 2015 at 12:19
  • $\begingroup$ @user1717828 khanacademy.org/math/algebra/… $\endgroup$ Jun 3, 2015 at 13:06
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    $\begingroup$ Worth noting, since the 'overdot' notation being standard in UK and apparently China has come up in comments on OP, that in that variant we dot the first and last in a sequence (as opposed to each digit). $\endgroup$
    – OJFord
    Jun 3, 2015 at 22:13
  • $\begingroup$ @user1717828 totally random, I didn't feel like going to the trouble of looking up a "significant" fraction. $\endgroup$
    – David Z
    Jun 4, 2015 at 5:08
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$0.\overline{9}=0.999999\ldots=1$

More generally, $0.\overline{n}=0.nnnnnnnnn\ldots$

For example, $\frac 13=0.333333333\ldots=0.\overline3$

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This symbol is called a vinculum or a overline.

The number(s) under this symbol are repeated indefinitely.

For your question:

$0.\overline{9} = 0.99999999999999...$

It could also be used like:

$2.52\overline{346} = 2.52346346346346...$

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  • $\begingroup$ What does this add to the existing, year-and-a-half-old answers? $\endgroup$ Dec 14, 2016 at 16:14
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    $\begingroup$ I am new, so I am just using this question to get my bearings on formating. $\endgroup$
    – Orange_Web
    Dec 14, 2016 at 16:18
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    $\begingroup$ Ah. I think this would be a better place to do that. Also, note that you don't actually have to submit your answer (or question) to see how it will format - this site does as-you-type previews (scroll down from the box where you're typing your answer/question; it may take a few seconds for the preview to form). One reason not to answer old questions to test formatting, is that any change to a question bumps it to the front page. $\endgroup$ Dec 14, 2016 at 16:25

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